Question

The production function of an isolated island economy is F(K; L) = 4K1=4L 3=4 .

(a) Assume the supply of labor is 1,296

(i) How much is the supply of capital if the market clears at a real rental rate of 8?

(ii) A hurricane hits the island in question. There are no casualties, but some capital stock has been destroyed. Would the equilibrium rental increase or decrease? Explain.

(b) Assume the supply of labor is 1,000. How much is the supply of capital if the market for LABOR clears at a real wage of 6?

Answer 1

Q = F(K, L) = 4K^1/4L^3/4

(a) L = 1,296

Q = 4 x K^1/4 x (1,296)^3/4 = 4 x 216 x K^1/4 = 864 x K^1/4

(i) Market will clear when MPK = Rental rate

MPK = dQ / dK = 864 x (1/4) / K^3/4 = 216 / K^3/4 = 8

K^3/4 = 216 / 8 = 27

Taking (4/3)rd root on each side,

K = 81

(ii) When capital stock is destroyed, supply of capital will fall, therefore rental rate for capital will increase ceteris paribus.

(b) L = 1,000

Market will clear when MPL = Wage rate

MPL = Q / L = 4 x (K / L)^1/4 = 4 x (K / 1,000)^1/4 = 4 x K^1/4 / 5.62 = 0.71 x K^1/4 = 6 (Wage rate)

K^1/4 = 0.71 / 6 = 0.1186

Raising each side to 4th power,

K = 0.0002